[Kadir Altintas] (*)
Denote:
A circle (A', x) intersects BC at Bc, Cb near to B, C, resp.
A circle (B', y) intersects CA at Ca, Ac near to C, A, resp.
A circle (C', z) intersects AB at Ab, Ba near to A, B, resp.
M1, M2, M3 = the midpoints of AbAc, BcBa, CaCb, resp.
1. ABC, M1M2M3 are perspective.
1. ABC, M1M2M3 are perspective.
2. ABC, M1M2M3 share the same centroid
3. Ab, Ac, Bc, Ba, Ca, Cb lie on a conic.
(*) Romantics of Geometry 1520
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If x, y, z (or a/2 -x =: X, b/2 -y =: Y, c/2 -z =: Z) are symmetric functions of a,b,c ie f(a,b,c), f(b,c,a), f(c,a,b) then the point of concurrence is a triangle center.
Which is it if x, y, z = exradii r_a, r_b, r_c, resp. or altitudes h_a, h_b, h_c, resp. ?
αph
If x, y, z (or a/2 -x =: X, b/2 -y =: Y, c/2 -z =: Z) are symmetric functions of a,b,c ie f(a,b,c), f(b,c,a), f(c,a,b) then the point of concurrence is a triangle center.
Which is it if x, y, z = exradii r_a, r_b, r_c, resp. or altitudes h_a, h_b, h_c, resp. ?
αph
[Angel Montesdeoca]:
Let ABC be a triangle.
1. r_a, r_b, r_c the exradii.
The circle B(r_a) of center B and radius r_a, cut BC into Ab and A'b.
The circle C(r_a) of center C and radius r_a, cut BC into Ac and A'c.
The points Bc, B'c, Ca, C'a, Ba, B'a, Cb, C'b are defined cyclically.
*** M1, M2, M3 = the midpoints of BcCb, CaAc y AbBa, resp.
The triangles ABC, M1M2M3 are perspective, with perspector
U = ( a(b+c-a)/(a(b+c-a) - S) : ... : ...),
lies on lines X(i)X(j) for these {i, j}: {1, 8958}, {9, 13388}, {165, 6213}, {223, 1212}, {281, 1659};
and (6 - 9 - 13) - search numbers (5.43524811180364, 3.87468486200098, -1.55038570492567).
*** M'1, M'2, M'3 = the midpoints of B'cC'b, C'aA'c y A'bB'a, resp.
The triangles ABC, M'1M'2M'3 are perspective, with perspector
V = ( a(b+c-a)/(a(b+c-a)+ S) : ... : ...),
lies on lines X(i)X(j) for these {i, j}: {9, 13389}, {165, 6212}, {223, 1212}, {281, 3536}, {558, 5451};
and (6 - 9 - 13) - search numbers (3.14538773488534, 2.05175767847960, 0.768499442243731).
*** N1, N2, N3 = the midpoints of BaCa, CbAb y AcBc, resp.
The triangles ABC, N1N2N3 are perspective, with perspector X(9).
*** N'1, N'2, N'3 = the midpoints of B'aC'a, C'bA'b y A'cB'c, resp.
The triangles ABC, N'1N'2N'3 are perspective, with perspector X(9).
*** The triangles M1M2M3, N1N2N3 are perspective, with perspector
Z = ((r+4R)^2-s(2r+12R-s)) X(6666) + (R s) X(13359) =
( 2a^6-3a^5(b+c)-3a^4(b+c)^2 +2a^3(3b^3+5b^2c+5b c^2+3c^3) +8a^2b c(b-c)^2 -a(b-c)^2(3b^3+13b^2c+13b c^2+3c^3) +(b-c)^4(b+c)^2
+ 2S(3a^3(b+c)-2a^2(3b^2+b c+3c^2) + 3a(b-c)^2(b+c)+2b c(b-c)^2 ) : ... : ...),
lies on lines X(i)X(j) for these {i, j}: {2, 13436}, {6666, 13359};
and (6 - 9 - 13) - search numbers (4.69425637776697, 0.824161421230953, 0.903511323624350).
*** The triangles M'1M'2M'3, N'1N'2N'3 are perspective, with perspector
W = ((r+4R)^2+s(2r+12R+s)) X6666 - (R s) X13359 =
( 2a^6-3a^5(b+c)-3a^4(b+c)^2 +2a^3(3b^3+5b^2c+5b c^2+3c^3) +8a^2b c(b-c)^2 -a(b-c)^2(3b^3+13b^2c+13b c^2+3c^3) +(b-c)^4(b+c)^2
- 2S(3a^3(b+c)-2a^2(3b^2+b c+3c^2) + 3a(b-c)^2(b+c)+2b c(b-c)^2 ) : ... : ...),
lies on lines X(i)X(j) for these {i, j}: {2, 13453}, {6666, 13360};
and (6 - 9 - 13) - search numbers (2.79510334680824, 1.84582232018845, 1.07273979248088).
2. h_a, h_b, h_c are the altitudes.
The circle B(h_a) of center B and radius h_a, cut BC into Ab and A'b.
The circle C(h_a) of center C and radius h_a, cut BC into Ac and A'c.
The points Bc, B'c, Ca, C'a, Ba, B'a, Cb, C'b are defined cyclically.
*** M1, M2, M3 = the midpoints of BcCb, CaAc y AbBa, resp.
The triangles ABC, M1M2M3 are perspective, with perspector X(494).
*** M'1, M'2, M'3 = the midpoints of B'cC'b, C'aA'c y A'bB'a, resp.
The triangles ABC, M'1M'2M'3 are perspective, with perspector X(493).
*** N1, N2, N3 = the midpoints of BaCa, CbAb y AcBc, resp.
The triangles ABC, N1N2N3 are perspective, with perspector X(6).
*** N'1, N'2, N'3 = the midpoints of B'aC'a, C'bA'b y A'cB'c, resp.
The triangles ABC, N'1N'2N'3 are perspective, with perspector X(6).
*** The triangles M1M2M3, N1N2N3 are perspective, with perspector
U1 = ( 2a^6 -3a^4(b^2+c^2) -12a^2b^2c^2 +(b^2-c^2)^2(b^2+c^2)
+ 2S(3a^2(b^2+c^2)+2b^2c^2) : ... : ...),
lies on line X(2)X(589);
and (6 - 9 - 13) - search numbers (2.25537763508478, 3.14015087717229, 0.425770350749050).
*** The triangles M'1M'2M'3, N'1N'2N'3 are perspective, with perspector
U2 = ( 2a^6 -3a^4(b^2+c^2) -12a^2b^2c^2 +(b^2-c^2)^2(b^2+c^2)
- 2S(3a^2(b^2+c^2)+2b^2c^2) : ... : ...),
lies on line X(2)X(588);
and (6 - 9 - 13) - search numbers (2.36643510911517, 1.75473740795483, 1.33364545680865).
More details in: http://amontes.webs.ull.es/otrashtm/HGT2018.htm#HG110118
Angel Montesdeoca
1. r_a, r_b, r_c the exradii.
The circle B(r_a) of center B and radius r_a, cut BC into Ab and A'b.
The circle C(r_a) of center C and radius r_a, cut BC into Ac and A'c.
The points Bc, B'c, Ca, C'a, Ba, B'a, Cb, C'b are defined cyclically.
*** M1, M2, M3 = the midpoints of BcCb, CaAc y AbBa, resp.
The triangles ABC, M1M2M3 are perspective, with perspector
U = ( a(b+c-a)/(a(b+c-a) - S) : ... : ...),
lies on lines X(i)X(j) for these {i, j}: {1, 8958}, {9, 13388}, {165, 6213}, {223, 1212}, {281, 1659};
and (6 - 9 - 13) - search numbers (5.43524811180364, 3.87468486200098, -1.55038570492567).
*** M'1, M'2, M'3 = the midpoints of B'cC'b, C'aA'c y A'bB'a, resp.
The triangles ABC, M'1M'2M'3 are perspective, with perspector
V = ( a(b+c-a)/(a(b+c-a)+ S) : ... : ...),
lies on lines X(i)X(j) for these {i, j}: {9, 13389}, {165, 6212}, {223, 1212}, {281, 3536}, {558, 5451};
and (6 - 9 - 13) - search numbers (3.14538773488534, 2.05175767847960, 0.768499442243731).
*** N1, N2, N3 = the midpoints of BaCa, CbAb y AcBc, resp.
The triangles ABC, N1N2N3 are perspective, with perspector X(9).
*** N'1, N'2, N'3 = the midpoints of B'aC'a, C'bA'b y A'cB'c, resp.
The triangles ABC, N'1N'2N'3 are perspective, with perspector X(9).
*** The triangles M1M2M3, N1N2N3 are perspective, with perspector
Z = ((r+4R)^2-s(2r+12R-s)) X(6666) + (R s) X(13359) =
( 2a^6-3a^5(b+c)-3a^4(b+c)^2 +2a^3(3b^3+5b^2c+5b c^2+3c^3) +8a^2b c(b-c)^2 -a(b-c)^2(3b^3+13b^2c+13b c^2+3c^3) +(b-c)^4(b+c)^2
+ 2S(3a^3(b+c)-2a^2(3b^2+b c+3c^2) + 3a(b-c)^2(b+c)+2b c(b-c)^2 ) : ... : ...),
lies on lines X(i)X(j) for these {i, j}: {2, 13436}, {6666, 13359};
and (6 - 9 - 13) - search numbers (4.69425637776697, 0.824161421230953, 0.903511323624350).
*** The triangles M'1M'2M'3, N'1N'2N'3 are perspective, with perspector
W = ((r+4R)^2+s(2r+12R+s)) X6666 - (R s) X13359 =
( 2a^6-3a^5(b+c)-3a^4(b+c)^2 +2a^3(3b^3+5b^2c+5b c^2+3c^3) +8a^2b c(b-c)^2 -a(b-c)^2(3b^3+13b^2c+13b c^2+3c^3) +(b-c)^4(b+c)^2
- 2S(3a^3(b+c)-2a^2(3b^2+b c+3c^2) + 3a(b-c)^2(b+c)+2b c(b-c)^2 ) : ... : ...),
lies on lines X(i)X(j) for these {i, j}: {2, 13453}, {6666, 13360};
and (6 - 9 - 13) - search numbers (2.79510334680824, 1.84582232018845, 1.07273979248088).
2. h_a, h_b, h_c are the altitudes.
The circle B(h_a) of center B and radius h_a, cut BC into Ab and A'b.
The circle C(h_a) of center C and radius h_a, cut BC into Ac and A'c.
The points Bc, B'c, Ca, C'a, Ba, B'a, Cb, C'b are defined cyclically.
*** M1, M2, M3 = the midpoints of BcCb, CaAc y AbBa, resp.
The triangles ABC, M1M2M3 are perspective, with perspector X(494).
*** M'1, M'2, M'3 = the midpoints of B'cC'b, C'aA'c y A'bB'a, resp.
The triangles ABC, M'1M'2M'3 are perspective, with perspector X(493).
*** N1, N2, N3 = the midpoints of BaCa, CbAb y AcBc, resp.
The triangles ABC, N1N2N3 are perspective, with perspector X(6).
*** N'1, N'2, N'3 = the midpoints of B'aC'a, C'bA'b y A'cB'c, resp.
The triangles ABC, N'1N'2N'3 are perspective, with perspector X(6).
*** The triangles M1M2M3, N1N2N3 are perspective, with perspector
U1 = ( 2a^6 -3a^4(b^2+c^2) -12a^2b^2c^2 +(b^2-c^2)^2(b^2+c^2)
+ 2S(3a^2(b^2+c^2)+2b^2c^2) : ... : ...),
lies on line X(2)X(589);
and (6 - 9 - 13) - search numbers (2.25537763508478, 3.14015087717229, 0.425770350749050).
*** The triangles M'1M'2M'3, N'1N'2N'3 are perspective, with perspector
U2 = ( 2a^6 -3a^4(b^2+c^2) -12a^2b^2c^2 +(b^2-c^2)^2(b^2+c^2)
- 2S(3a^2(b^2+c^2)+2b^2c^2) : ... : ...),
lies on line X(2)X(588);
and (6 - 9 - 13) - search numbers (2.36643510911517, 1.75473740795483, 1.33364545680865).
More details in: http://amontes.webs.ull.es/otrashtm/HGT2018.htm#HG110118
Angel Montesdeoca
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